Localized shocks

Roberts, Daniel A.; Stanford, Douglas; Susskind, Leonard

doi:10.1007/jhep03(2015)051

Cited by 598 publications

(870 citation statements)

References 79 publications

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“…For example, a Pauli operator initially acting on a single spin (e.g., O i ≡ Y i ; we denote the Pauli matrices by X, Y, Z) will evolve with time into an operator O i ðtÞ which acts on many spins. Operators typically grow ballistically [38], in the sense that the number of spins in the support of O i ðtÞ grows linearly with t. In this section, we relate the growth of the bipartite entanglement to the spreading of operators. We focus on the special case of unitary evolution with Clifford circuits (defined below), but we expect the basic outcomes to hold for more general unitary dynamics.…”

Section: Hydrodynamics Of Operator Spreadingmentioning

confidence: 99%

“…The entanglement entropy, and even its time dependence, is also beginning to be experimentally measurable in cold atom systems [32][33][34]. In a very different context, black holes have motivated studies of how fast quantum systems can scramble information by dynamically generating entanglement [35][36][37][38]. Simple quantum circuitsquantum evolutions in discrete time-serve as useful toy models for entanglement growth and scrambling [39][40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

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Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the "entanglement tsunami" in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated KardarParisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like ðtimeÞ 1=3 and are spatially correlated over a distance ∝ ðtimeÞ 2=3 . We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a "minimal cut" picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the "velocity" of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the wellstudied problem of pinning of a membrane or domain wall by disorder.

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Section: Hydrodynamics Of Operator Spreadingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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“…More generally, one would like to identify vB with a universally defined nonquasiparticle velocity. It has been noted (54) that, in some nonquasiparticle systems, it is the "butterfly velocity" (55,56) that appears in the diffusivity equation (Eq. 3).…”

Section: Significancementioning

confidence: 99%

Anomalous thermal diffusivity in underdoped YBa ₂ Cu ₃ O _6+x

Zhang

Levenson-Falk

Ramshaw

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

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The thermal diffusivity in the ab plane of underdoped YBCO crystals is measured by means of a local optical technique in the temperature range of 25-300 K. The phase delay between a point heat source and a set of detection points around it allows for high-resolution measurement of the thermal diffusivity and its in-plane anisotropy. Although the magnitude of the diffusivity may suggest that it originates from phonons, its anisotropy is comparable with reported values of the electrical resistivity anisotropy. Furthermore, the anisotropy drops sharply below the charge order transition, again similar to the electrical resistivity anisotropy. Both of these observations suggest that the thermal diffusivity has pronounced electronic as well as phononic character. At the same time, the small electrical and thermal conductivities at high temperatures imply that neither well-defined electron nor phonon quasiparticles are present in this material. We interpret our results through a strongly interacting incoherent electron-phonon "soup" picture characterized by a diffusion constant D ∼ v 2 B τ , where v B is the soup velocity, and scattering of both electrons and phonons saturates a quantum thermal relaxation time τ ∼ /k B T.thermal diffusivity | bad metals | electron-phonon T he standard paradigm for transport in metals relies on the existence of quasiparticles. Electronic quasiparticles conduct electricity and heat. Phonon quasiparticles, the collective excitations of the elastic solid (here, we discuss acoustic phonons) also conduct heat. Transport coefficients, such as electrical and thermal conductivities, can then be calculated using, for example, Boltzmann equations (1). However, such an approach fails when the quasiparticle mean free paths become comparable with the quasiparticle wavelength. For electrons, it is the Fermi wavelength (2-4), whereas for phonons, it is the larger of the interatomic distance or minimum excited phonon wavelength (5, 6). Understanding transport in nonquasiparticle regimes requires a new framework and has become a subject of intense theoretical effort in recent years, triggering an urgent need for experimental results that can shed light on such regimes. In particular, in ref. 7, the diffusivity was singled out as a key observable for incoherent nonquasiparticle transport, possibly subject to fundamental quantum mechanical bounds.In this work, we report high-resolution measurements of the thermal diffusivity of single-crystal underdoped YBCO6.60 (an ortho-II YBa2Cu3O6.60) and YBCO6.75 (an ortho-III YBa2Cu3O6.75). We are particularly interested in the anisotropy of the thermal diffusivity as measured along the principal axes a and b (which is the chain direction) in the temperature range 25-300 K. We use a noncontact optical microscope (see Fig. S1) to perform local thermal transport measurements on the scale of ∼10 µm, hence avoiding inhomogeneities, particularly twinning and grain boundary effects. Our principal experimental results are as follows. (i) The measured thermal diffusivity is...

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“…For more details, we refer to [15,19,[37][38][39][40][41][42][43]. The butterfly effect as chaotic behaviour refers to the exponential growth of a small perturbation to a quantum system.…”

Section: Butterfly Velocitymentioning

confidence: 99%

Diffusion and butterfly velocity at finite density

Niu

Kim

2017

J. High Energ. Phys.

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Abstract:We study diffusion and butterfly velocity (v B ) in two holographic models, linear axion and axion-dilaton model, with a momentum relaxation parameter (β) at finite density or chemical potential (µ). Axion-dilaton model is particularly interesting since it shows linear-T -resistivity, which may have something to do with the universal bound of diffusion. At finite density, there are two diffusion constants D ± describing the coupled diffusion of charge and energy. By computing D ± exactly, we find that in the incoherent regime (β/T 1, β/µ 1) D + is identified with the charge diffusion constant (D c ) and D − is identified with the energy diffusion constant (D e ). In the coherent regime, at very small density, D ± are 'maximally' mixed in the sense that D + (D − ) is identified with D e (D c ), which is opposite to the case in the incoherent regime. In the incoherent regime D e ∼ C − v 2 B /k B T where C − = 1/2 or 1 so it is universal independently of β and µ. However, D c ∼ C + v 2 B /k B T where C + = 1 or β 2 /16π 2 T 2 so, in general, C + may not saturate to the lower bound in the incoherent regime, which suggests that the characteristic velocity for charge diffusion may not be the butterfly velocity. We find that the finite density does not affect the diffusion property at zero density in the incoherent regime.

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Localized shocks

Cited by 598 publications

References 79 publications

Quantum Entanglement Growth under Random Unitary Dynamics

Quantum Entanglement Growth under Random Unitary Dynamics

Anomalous thermal diffusivity in underdoped YBa ₂ Cu ₃ O _6+x

Diffusion and butterfly velocity at finite density

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Localized shocks

Cited by 598 publications

References 79 publications

Quantum Entanglement Growth under Random Unitary Dynamics

Quantum Entanglement Growth under Random Unitary Dynamics

Anomalous thermal diffusivity in underdoped YBa 2 Cu 3 O 6+x

Diffusion and butterfly velocity at finite density

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Anomalous thermal diffusivity in underdoped YBa ₂ Cu ₃ O _6+x